Filippov showed interest in continuous loops in 1950 when he constructed a proof that they divide a plane into interior and exterior parts.[1] Known as the Jordan curve theorem, it exemplifies a mathematical proposition easily stated but difficult to prove.
In 1955 Filippov and V. S. Ryaben'kii became interested in difference equations and wrote On the Stability of Difference Equations.[2] The work was developed into a textbook in 1961 which was used in Moscow State University and many other Russian universities for several decades.[3]
In 1959 he published a paper containing a lemma about implicit functions designed for use in optimal control theory that is named after him (Filippov's lemma).[4][5][6]
Filippov made an important contribution in the theory of discontinuous ordinary differential equations with his monograph Differential Equations with Discontinuous Righthand Sides (1985).[7] Such set-valued dynamical systems arise in sliding mode control, an important class of feedback control systems demonstrating robust control. Such systems model some mechanical systems with Coulomb friction, and more recently genetic networks.
^S. Nababan (1979) "A Filippov-type lemma for functions involving delays and its application to time-delayed optimal control problems", Journal of Optimization Theory and Applications 27(3):357–76
